{"id":"cb_question_bank:bb882bdf-8b6b-46f5-bc13-213abec5752d","source_id":"cb_question_bank","source_item_id":"bb882bdf-8b6b-46f5-bc13-213abec5752d","external_id":null,"ibn":"025860-DC","question_id":"12e7faf8","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.A.","skill":"Equivalent expressions","difficulty":"H","score_band":6,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"The equation the fraction with numerator x squared, plus 6 x, minus 7, and denominator x plus 7, end fraction, equals, a x plus d is true for all x not equal to negative 7 , where a and d are integers. What is the value of a plus d ?","stimulus":"","stimulus_html":null,"stem_html":"<p class=\"stem_paragraph \">The equation <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/cb5e3b8df5ce10d0bbc2c2bdf5e966acefd75bb16c1a0645bd351359d69f4b0c\" alt=\"the fraction with numerator x squared, plus 6 x, minus 7, and denominator x plus 7, end fraction, equals, a x plus d\"></span></span> is true for all <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/86c9818dcd419bda0e9e7ee3c44568d7077c7650d4856c9437da7dd26d348879\" alt=\"x not equal to negative 7\"></span></span>, where <span class=\"italic\">a</span> and <span class=\"italic\">d</span> are integers. What is the value of <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/3a10840f8531bd89c2cce367998f7042939ce51dca56240fa34e9b3c7e2bec4c\" alt=\"a plus d\"></span></span> ?</p>\n","rationale_html":"<p>Choice C is correct. Since the expression <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/3262364d582d610724bc7cc095e6924b1d9b8ee2c45a6247d84d8c7d5ec5f881\" alt=\"x squared, plus 6 x, minus 7\"></span> can be factored as <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/2edfe85166b940a68965323db9df1995c49850ad7423acab5d40cbc4fbb6bf48\" alt=\"open parenthesis, x plus 7, close parenthesis, times, open parenthesis, x minus 1, close parenthesis\"></span><span class=\"italic\"></span><span class=\"italic\"></span>, the given equation can be rewritten as <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/91bee89d642deebfc0a58d120244c239c9c22ad659f8b4b5072697a041e08917\" alt=\"the fraction with numerator open parenthesis, x plus 7, close parenthesis, times, open parenthesis, x minus 1, close parenthesis, and denominator x plus 7, end fraction, equals a, x plus d\"></span>. Since <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/86c9818dcd419bda0e9e7ee3c44568d7077c7650d4856c9437da7dd26d348879\" alt=\"x is not equal to negative 7\"></span>, <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/efbc83d48599f9fc1ebeab42dd3c4162037a5f3c29b926b2c3f67a15971cdc4e\" alt=\"x plus 7\"></span> is also not equal to 0, so both the numerator and denominator of <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/07e8fe542db6a36fde40ed3ed41ffc5207b59afcc605a68ce50e958481781c1d\" alt=\"the fraction with numerator open parenthesis, x plus 7, close parenthesis, times, open parenthesis, x minus 1, close parenthesis, and denominator x plus 7, end fraction\"></span> can be divided by <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/efbc83d48599f9fc1ebeab42dd3c4162037a5f3c29b926b2c3f67a15971cdc4e\" alt=\"x plus 7\"></span>. This gives <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/a086f22733f367cb3c8d594d6d4e75aae85cfcb18d72875a808e8ba00ae889db\" alt=\"x minus 1, equals a x plus d\"></span>. Equating the coefficient of <span class=\"italic\">x</span> on each side of the equation gives <span class=\"italic\"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/35066e9ba042cc64fd96f387da862ae288c94643a339ae4d2b59cd378d89d3a4\" alt=\"a, equals 1\"></span></span>. Equating the constant terms gives <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/54774fda56e9ae3cbf59b20ee9036305f56c8116e6dd7ad363641b4e128aa601\" alt=\"d equals negative 1\"></span>. The sum is <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/cf6e9f36787eefeba9d592a2e19b0a4250cc6b89ccbefd95520882c606e2010e\" alt=\"1 plus negative 1, equals 0\"></span>.</p><p>Choice A is incorrect and may result from incorrectly simplifying the equation. Choices B and D are incorrect. They are the values of <span class=\"italic\">d </span>and <span class=\"italic\">a</span>, respectively, not <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/3a10840f8531bd89c2cce367998f7042939ce51dca56240fa34e9b3c7e2bec4c\" alt=\"a, plus d\"></span>.</p><p></p>\n","correct_answer":["C"],"has_media":true,"has_mathml":false,"has_table":false,"source_created_at":"2023-08-02T20:25:59.620000+00:00","source_updated_at":"2023-08-02T20:25:59.620000+00:00","options":[{"label":"A","content_html":"<span class=\"choice_cell align:right \">\n            <span class=\"math_expression \">\n              <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/f8ac4e862b233acbee95c501658ff49dd21661d2a931c76a597fe7d20c5d70df\" alt=\"negative 6\"></span></span>\n          </span>\n","ord":0},{"label":"B","content_html":"<span class=\"choice_cell align:right \">\n            <span class=\"math_expression \">\n              <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/b3ef0bab498e581649a76baff7488f24fc7f565702b5178801873c0eccc8d6a5\" alt=\"negative 1\"></span></span>\n          </span>\n","ord":1},{"label":"C","content_html":"<span class=\"choice_cell align:right \">\n            <span class=\"math_expression \">\n              <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/e646b570dc3598aa1fe55743f6c749ef456a13deb8575aca77a309272b8cbc26\" alt=\"0\"></span></span>\n          </span>\n","ord":2},{"label":"D","content_html":"<span class=\"choice_cell align:right \">\n            <span class=\"math_expression \">\n              <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/2f1494c799ca1decbcb94ac7e70aeb91da2c02dfba7cb54dae8e89aee537a8e1\" alt=\"1\"></span></span>\n          </span>\n","ord":3}],"attribution":{"source_id":"cb_question_bank","source_name":"College Board digital SAT question bank (community dump)","rights_holder":"College Board","canonical_url":"https://satsuitequestionbank.collegeboard.org/","retrieved_from":"https://raw.githubusercontent.com/mdn522/sat-question-bank/main/data/cb-digital-questions.json","attribution":"SAT practice questions © College Board, from the official SAT Suite Question Bank. Retrieved via the community dump at github.com/mdn522/sat-question-bank.","disclaimer":"SAT® and PSAT/NMSQT® are trademarks registered by the College Board, which is not affiliated with, does not sponsor, and does not endorse this project.","license":"College Board copyright; source repository carries no licence. Local use only.","redistributable":false,"item_count":2017}}